The Teichmueller space of a surface S parametrizes hyperbolic structures on S; it can be embedded as a connected component of the representation variety of the fundamental group of S into PSL(2,R). Higher Teichmueller spaces are connected components of representation varieties of the fundamental group of S into more general Lie groups G, which share several algebraic and geometric properties with classical Teichmueller space. A central question is if higher Teichmueller spaces also parametrize geometric structures on the surface S.